I saw a notice in a local restaurant,
"we accept credit card payment".
I am wondering if it would be better if the restaurant wrote,
we accept payment by credit card.
The two sentences that you present are entirely synonymous. In this case, there is no difference in their meaning, both connotatively as well as denotatively.
As Damkerng said in the comments, the best option in my opinion (and the most common in my experience) is
We accept credit cards.
The fact that the credit cards are used for payment doesn't need to be stated - it is implied, because that is the only thing you can use credit cards for.